# NegativelyOrientedPoints

NegativelyOrientedPoints[{p1,p2,p3,,pn}]

tests whether the sequence of points p1,p2,p3,,pn is negatively oriented.

# Details • NegativelyOrientedPoints is also known as clockwise in 2D and lefthand rule in 3D.
• Typically used to determine the orientation of a rotational motion with respect to a set of points.
• • In two dimensions, NegativelyOrientedPoints[{p1,p2,p3}] gives True if the point p3 is in the half-plane bounded by the line through p1 and p2 and extended in the direction of {1,0}.
• For negatively oriented points p1, p2 and p3, the determinant of the matrix {p2-p1, p3-p1} is negative.
• • In three dimensions, NegativelyOrientedPoints[{p1,p2,p3,p4}] gives True if the point p4 is in the half-space bounded by the plane through the point p1 with normal direction (p3-p1)(p2-p1).
• For negatively oriented points p1, p2, p3 and p4, the dot product of p4-p1 and (p3-p1)(p2-p1) is negative.
• In d dimensions, d+1 points p1,p2,,pd+1 are negatively oriented if the determinant of the matrix {p2-p1,,pd+1-p1} is negative.

# Examples

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## Basic Examples(2)

The points {0,0}, {1,1}, {.5,-1} are negatively oriented:

Plot the points:

Find the condition for which a point is below a plane:

## Scope(3)

NegativelyOrientedPoints works with two-dimensional points:

Three-dimensional points:

NegativelyOrientedPoints works with numerical coordinates:

Symbolic coordinates:

NegativelyOrientedPoints over a set of coordinates:

List of points:

Multi-points:

## Generalizations & Extensions(1)

Give assumptions to NegativelyOrientedPoints:

## Applications(5)

### Basic Applications(2)

Graph negatively oriented points:

Show the left-hand rule:

### Geometry(3)

Faces of a polyhedron are positively oriented:

NegativelyOrientedPoints over lines in 2D:

It is equivalent to the orientation of the consecutive vertices of the line:

Show the robustness of NegativelyOrientedPoints:

## Properties & Relations(4)

NegativelyOrientedPoints returns False for collinear points:

NegativelyOrientedPoints returns False if positively oriented:

Use RegionMember to test whether points are negatively oriented:

3D points are coplanar if they are neither positively nor negatively oriented: